Closed-Loop Data Transcription to an LDR via Minimaxing Rate Reduction

This work proposes a new computational framework for learning an explicit generative model for real-world datasets. In particular we propose to learn {\em a closed-loop transcription} between a multi-class multi-dimensional data distribution and a { linear discriminative representation (LDR)} in the feature space that consists of multiple independent multi-dimensional linear subspaces. In particular, we argue that the optimal encoding and decoding mappings sought can be formulated as the equilibrium point of a {\em two-player minimax game between the encoder and decoder}. A natural utility function for this game is the so-called {\em rate reduction}, a simple information-theoretic measure for distances between mixtures of subspace-like Gaussians in the feature space. Our formulation draws inspiration from closed-loop error feedback from control systems and avoids expensive evaluating and minimizing approximated distances between arbitrary distributions in either the data space or the feature space. To a large extent, this new formulation unifies the concepts and benefits of Auto-Encoding and GAN and naturally extends them to the settings of learning a {\em both discriminative and generative} representation for multi-class and multi-dimensional real-world data. Our extensive experiments on many benchmark imagery datasets demonstrate tremendous potential of this new closed-loop formulation: under fair comparison, visual quality of the learned decoder and classification performance of the encoder is competitive and often better than existing methods based on GAN, VAE, or a combination of both. We notice that the so learned features of different classes are explicitly mapped onto approximately {\em independent principal subspaces} in the feature space; and diverse visual attributes within each class are modeled by the {\em independent principal components} within each subspace.